INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 15 (2003) S2533–S2546 PII: S0953-8984(03)62916-4

Magnetism in monatomic metal wires

P Gambardella

Institut de Physique des Nanostructures, Ecole Polytechnique Federale de Lausanne,CH-1015 Lausanne, Switzerland

E-mail: pietro.gambardella@epfl.ch

Received 30 April 2003Published 15 August 2003Online at stacks.iop.org/JPhysCM/15/S2533

AbstractThe first investigation of magnetism in one-dimensional (1D) monatomic chainsof metal atoms is reported. High-density arrays (5 × 106 cm−1) of parallelCo atomic wires have been grown using the vicinal Pt(997) surface as atemplate. Angle-resolved photoemission experiments evidence the presenceof a 3d exchange-split band for the Co wires giving rise to enhanced localizedspin magnetic moments. X-ray magnetic circular dichroism shows furtherthat the orbital magnetic moment is about five times larger compared to thatof bulk hcp Co as a result of the reduced atomic coordination of the 1Dwires. Whereas statistical models forbid long-range ferromagnetic order ininfinite 1D spin chains at any temperature greater than zero, we show that finitemonatomic Co chains display both short- and long-range ferromagnetic order.The chains consist of thermally fluctuating segments of ferromagneticallycoupled atoms which evolve into a ferromagnetic metastable long-range-ordered state below 15 K. Ferromagnetism in 1D is stabilized by unusuallylarge magnetic anisotropy energy barriers (2 meV/atom) which arise from thereduced dimensionality of the wires and related large orbital magnetization.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Fundamental magnetic properties such as spontaneous magnetization, magnetocrystallineanisotropy, and magnitudes of the spin (S) and orbital (L) atomic magnetic moments dependinherently on the dimensionality of a given system [1–5]. According to statistical mechanics,the tendency to magnetic order reduces as thermal fluctuations become more disruptive withdecreasing dimensionality. As a result, magnetism in two-dimensional (2D) ultrathin films ismore sensitive to temperature effects compared to that in 3D systems [6]. Indeed, owing to thereduced number of neighbours contributing to the exchange interaction, rigorous results for theisotropic Heisenberg model rule out ferromagnetic, antiferromagnetic, as well as oscillatory

0953-8984/03/342533+14$30.00 © 2003 IOP Publishing Ltd Printed in the UK S2533

S2534 P Gambardella

long-range order in 2D and 1D [7, 8]. However, in 2D the introduction of dipolar couplingor of an arbitrarily small anisotropy of spin–orbit origin is sufficient to establish long-rangemagnetic order [9], while in 1D the Ising model, i.e., the extreme anisotropic limit of theHeisenberg Hamiltonian, still predicts zero magnetization in the absence of an external fieldat T > 0. Such conclusions apply to an ideal 1D lattice of spin point vectors. Linear chains ofreal atoms, however, differ substantially from a spin lattice model in that they have finite length,and S (L) arise from 3D spin (charge) densities distributed across the unit cell. Magnetic orderin 1D systems consisting of real atoms is therefore an unsettled issue.

A further topic of fundamental importance is how S, L, and the magnetic anisotropyenergy per atom, Ea , of a 1D system change with respect to those of the correspondingbulk elements. It is known that the reduced atomic coordination in thin films [1, 10–12]and small particles [4, 13, 14] significantly alters such quantities. In 1D, depending on thesubstrate chosen as a support, even stronger changes in S, L, and Ea are expected [15–23].Theoretical models [24, 25] show that the magnetocrystalline anisotropy is connected via Land the spin–orbit interaction to the atomic structure of a magnetic material. There is thereforethe possibility of enhancing the magnetic anisotropy by artificially decreasing the symmetryand coordination of magnetic aggregates, as is the case for 1D atomic chains. Theoreticalcalculations [16, 18, 19] show that values of Ea of the order of 10−3–10−2 eV/atom can beattained, a factor 103 larger compared to those for bulk ferromagnetic metals.

Most experimental investigations on 1D magnetic systems have concentrated oninsulator crystals consisting of arrays of linear chains of exchange-coupled transition metal(TM) ions separated by non-magnetic atom spacers and characterized by weak interchaininteractions [26, 27]. One example is tetramethyl ammonium manganese chloride, whichtypically exhibits antiferromagnetic coupling and 1D paramagnetic behaviour down to aboutT = 1 K [28]. More recently, thanks to progress in molecular beam epitaxy at atomicallyordered surfaces, investigations have focused on pure metal nanowires, whose thickness andspatial separation can be independently controlled. The atomic steps of a non-magnetic vicinalsurface are used as a deposition template for TM atoms, thus producing a large number ofnanowires in a parallel process. A small cost is paid in terms of the finite-size distributionof the wires with respect to, e.g., atom manipulation by scanning probes [29], but the largewire density allows one to use spatially integrating techniques with magnetic sensitivity suchas Kerr magnetometry and x-ray magnetic circular dichroism (XMCD). This approach wasfirst explored by Elmers et al [30] in the study of Fe wires on vicinal W(110), which showin-plane anisotropy and a relaxation-free ferromagnetic phase transition due to dipole-inducedcoupling across adjacent stripes [31, 32]. Shen et al [33, 34] found a pronounced temperature-and time-dependent magnetic relaxation for Fe stripes on stepped Cu(111) with out-of-planeanisotropy, due to the formation of 1D Ising-coupled spin blocks. Fe and Co stripes on Pd(110)and Ru(0001), respectively, have also been studied [35, 36].

In this study, we present the first experimental investigation of the magnetism of 1D metalchains in the monatomic limit. In section 2, we present the self-assembly approach that allowsone to grow uniform arrays of 1D wires on a Pt stepped surface. We address the magneticproperties of 1D Co chains by analysing their valence band photoemission spectra (section 3)and XMCD data (section 4).

2. Self-assembly of Co monatomic chains

In this section we address the nucleation and growth of metals on densely stepped substrateswith the aim of creating arrays of 1D nanowires with precise morphological characteristics.Depending on the surface temperature, adatoms on vicinal surfaces self-assemble into chain-

Magnetism in monatomic metal wires S2535

(i) ideal row-by-row growth

(iii) rough growth (v) double wires

(ii) non-periodic pattern

(iv) alloying

slow edgediffusion

fast edge diffusion

interlayer mass transport

alloying

100 200 300 400 500 600 T (K)

wire growthperiodicarrays

rough growth

(a)

(b)

Figure 1. (a) Different growth modes on a stepped substrate. (b) Co growth modes on Pt(997) asa function of the substrate temperature. Although the temperature scale refers to the Co/Pt(997)system, this description applies to other metals such as Ag and Cu.

like structures by decorating the step edges. This is simply due to the increase of binding energyat the step sites [37, 38] and is, of course, a general phenomenon that holds for metals [39–44]as well as for gas species [45, 46]. An advantage of this growth method is that by adjusting theadatom coverage and the average step spacing one can independently control the wire width andseparation, respectively. Growth proceeds either as a smooth step-wetting process [40–43] oras nucleation of two-dimensional (2D) islands at the step edges [47], provided that the adatomdeplacement prior to nucleation is larger than the terrace width of the substrate. In figure 1we show different scenarios of heteroepitaxy on a stepped substrate. We distinguish (i) theideal case of row-by-row growth, (ii) wires of different widths due to interlayer crossing of theadatoms, (iii) formation of irregular 2D islands at the step edges, (iv) alloying, (v) formation ofdouble-layer wires. In preparing arrays of 1D wires for photoemission and magnetic dichroismexperiments, we have focused on the conditions that favour the ideal case (i). As a general trendwe find that wire formation is limited at low temperature by slow edge-diffusion processesand at high temperature by interlayer diffusion and, eventually, by alloying between the metaladspecies and the substrate.

The basic requirement for growing self-assembled patterns of regular wires by stepdecoration is a good template. By this we mean a sample whose steps are as straight andas evenly spaced as possible. We have chosen to work with Pt vicinal surfaces since repulsiveinteractions between adjacent steps suppress step meandering [48], resulting in remarkablystraight steps and in a narrow terrace width distribution, as shown in the scanning tunnellingmicroscopy (STM) image in figure 2(a). The average step separation as well as the kink densityalong the steps are determined by the crystal miscut. In the Pt(997) case, the average terracewidth is 20.2 Å (figure 2(b)) with standard deviation σ = 2.9 Å [49], which allows one toobtain a density of 5×106 atomic wires cm−1.

S2536 P Gambardella

a) c)b)

d)

200 A 100 A

Figure 2. STM topographs of the Pt(997) surface. (a) Periodic step structure (each white linerepresents a monatomic Pt step). The step-down direction is from right to left. (b) A 3D close-upof the Pt steps (the vertical scale has been enhanced for better rendering). (c) Pt(997) after 10 minof sputtering at 300 K with a 800 eV Ar+ beam at normal incidence. (d) Pt(997) after annealing inthe presence of contaminants.

The cleaning of the sample is important for obtaining a regular periodic substrate. Aftersputtering at 300 K with a 800 eV Ar+ beam, the stepped structure of the surface is completelylost (figure 2(c)). It can be recovered by annealing the surface at high temperature. To avoiduneven removal of material from the surface, however, the sample temperature is usually keptat T = 750 K during sputtering to allow sufficient mobility of the Pt atoms, and the ionbeam is directed either normal to the surface or parallel to the steps. After repeated cycles of800 eV Ar+ sputtering at 750 K, the surface is annealed to 850 K, followed by a brief exposureto 1 × 10−7 mbar oxygen and by a flash to T > 1000 K to remove residual contaminant. Carehas to be taken in cooling the sample at a slow enough rate (<40 K min−1) down to 500 K inorder to allow equilibration of the step morphology. Annealing in the presence of impuritiesmight result in step pinning during the cool down of the surface (figure 2(d)).

Single Co atoms on Pt(111) terraces are mobile above T = 55 K. At higher temperature, asthe terrace width of Pt(997) is small compared to the mean free path of Co adatoms, nucleationat the step sites occurs. The wire growth proceeds via incorporation of adatoms in 1D stablenuclei attached to the step edges [42]. However, below 250 K the wire formation is kineticallyhindered by slow edge- and corner-diffusion processes. Regular Co wires grow only above250 K [43], as shown in figure 3. A monatomic chain array is obtained as the coverage equalsthe inverse of the number of atomic rows in the substrate terraces, i.e., 0.13 monolayer (ML)(1 ML = 1.5 × 10−15 atoms cm−2) for Pt(997). The average length of a continuous Co chainis estimated to be about 80 atoms from the average kink density per Pt step. As the coverageincreases to more than a single monatomic wire per terrace, Co grows row by row (figures 4(a),(b)) up to T = 290 K. At T > 290 K, interlayer diffusion, i.e., the diffusion of adatoms acrossadjacent terraces, sets in. This implies that the proportionality between adlayer coverage oneach terrace and terrace width is no longer valid, resulting in either or both (ii) and (v), shownin figure 1(a). As Co adatoms acquire enough thermal energy to cross the Pt–Co boundary atsteps, bilayer Co wires and step bunching are observed (figure 4(c)).

Below 1 ML, the Co wires are pseudomorphic with the Pt substrate, implying that the Colattice parameter is expanded by about 10% with respect to that of bulk hcp Co. In contrast tothe case for deposition of Co on Pt(111) [50, 51], we do not observe a reconstruction of the

Magnetism in monatomic metal wires S2537

Co atomsPt terrace x

z

y

Figure 3. Co monatomic chains decorate the Pt step edges following deposition of 0.07 ML Coat T = 250 K (the vertical scale has been enhanced for better rendering). The chains are linearlyaligned and have a spacing equal to the terrace width. The protrusion on the terrace is attributed toCo atoms incorporated in the Pt layer (from [67]).

CoCo

PtPt

b)

a)

50 A

c)

50 A

Co PtPtCo

Figure 4. An STM image of 0.6 ML Co deposited at T = 250 K; the step-down direction isfrom right to left. Row-by-row growth conserves the original step pattern, forming regular stripesthat run parallel to the Pt steps. (b) Detail of two adjacent terraces, as shown in the diagram, withatomic resolution on the Co chains and Pt substrate. (c) An STM image of 0.6 ML Co deposited atT = 385 K; the step-down direction is as in (a). Formation of bilayer Co wires with a non-periodicpattern.

substrate and the formation of dendrite-like islands, although stacking faults appear alreadyat 250 K at the completion of the first Co overlayer [43]. At T > 500 K, significant Co–Ptalloying takes place. The temperature window for the growth of a regular periodic array ofCo atomic chains is therefore limited to a narrow interval between 250 and 300 K, as edgediffusion is fast enough to have row-by-row growth and interlayer diffusion and intermixingare still inhibited (figure 1(b)).

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20 A

Figure 5. An STM image of 0.2 ML Ni deposited on Pt(997) at 200 K. The arrows indicate Niatoms that have exchanged with Pt at step and terrace sites. The step-down direction is from rightto left.

Not all combinations of substrate and overlayer are viable for inducing the self-assemblyof monatomic chains. Ni, in contrast to Co, Ag, and Cu [42], is already diffusing into the Pttopmost layer at 150 K [44]. The adislands at this temperature have a rough 2D character;it is therefore impossible to find a temperature range where adatom diffusion is fast enoughto lead to row-by-row growth prior to alloying. In the STM images, Ni atoms are imaged asbright protrusions (figure 5) along the step edges and on the terraces. Intermixing proceedsmainly via the substrate steps. Pt atoms that have exchanged with Ni are incorporated intothe islands that decorate the step edges. A 2 × 1 Ni superstructure with 5.5 Å periodicity,i.e., two Pt lattice constants, forms confined to the step edges. This phase can be consideredthe 1D analogue of the NiPt3 bulk alloy, where Ni atoms have only Pt nearest neighbours. Themagnetic properties of such a 1D alloy have not been investigated here.

3. Electronic states of monatomic Co and Cu chains

The physical realization of 1D atomic systems opens up the possibility of investigatingtheir magnetic behaviour and related electronic structure. Angle-resolved photoemissionexperiments have been performed at BESSY I in Berlin, at the TGM 5 undulator beamline.Considering the small coverage of Co that corresponds to a monatomic chain array on Pt(997),a serious problem encountered in the experiment was the isolation of the Co chain-inducedstates from the Pt 5d background in angle-resolved photoemission spectra. In order to increasethe sensitivity to Co over Pt, we have exploited the difference in photon energy dependence ofthe 3d and 5d photoionization cross-sections. At a photon energy hν = 40 eV the spectra ofclean Pt(997) and of the Co monatomic chains (0.12 ML Co) are nearly the same (figure 6(a),top). The presence of the Cooper minimum [52] in the Pt 5d cross-section at hν = 150 eV,however, significantly reduces the contribution of the Pt states for hν > 100 eV, so states withprevalent 3d character can be identified in the photoemission spectra (figure 6(a), bottom).

Figure 6(b) shows the evolution of the Co 3d states with Co coverage. Two featuresappear for the monatomic chains: one close to the Fermi level at about 0.3 eV and theother approximately at 2.4 eV binding energy. These features become more pronouncedwith increasing coverage, while their separation decreases. The observation of a double-peaked structure for the Co wires suggests the presence of a 1D exchange-split Co band [52].

Magnetism in monatomic metal wires S2539

Pt0.010.030.070.090.120.150.170.210.260.300.400.500.751.001.502.00

Cu (ML)

0-1-2-3-4-5-6

Binding energy (eV)

Pt

0.130.150.180.210.240.270.300.330.40

0.10

Co (ML)

Binding energy (eV)

0-1-2-3-4-5-6

b) c)

5 4 3 2 1 0

0.12 ML Co/Pt(997)

Pt(997)

hν = 40 eV

hν = 143 eV

Binding energy (eV)

Inte

nsity

(ar

b. u

nits

)a)

Figure 6. (a) Photoemission spectra of clean Pt(997) and after the deposition of 0.12 ML Co takenat a photon energy hν = 40 (top) and 143 eV (bottom). (b) Photoemission spectra of Co/Pt(997)taken at hν = 122 eV, 4◦ off normal emission, displaying the evolution of Co states with increasingCo coverage. The solid lines indicate the exchange-split Co 3d states. (c) Photoemission spectraof Cu/Pt(997) taken at hν = 122 eV at normal emission displaying the evolution of Cu states onPt(997). A single peak is observed for the non-magnetic Cu d states (from [52]).

Exchange splitting is one key to magnetism since it creates a spin imbalance that produces alocal magnetic moment. An approximate linear correlation between the 3d magnetic splittingand the local magnetic moment per atom has been found for TMs, where the 3d moment isof the order of 1 µB eV−1 times the exchange splitting [3]. For the Co monatomic chain-induced states, the exchange splitting is large (≈2.1 eV) compared to typical values for Cothin films (1.4–1.9 eV) and bulk Co (≈1.4 eV) [53–55]. This suggests in turn that S isconsiderably enhanced in the Co monatomic chains, of the order of 2.1 µB , in agreement withcalculations [15, 17–19, 56]. Due to the 3d band broadening, the magnitude of the exchangesplitting reduces with increased Co atomic coordination, as seen in figure 6(b) for coverageslarger than 0.13 ML. In contrast to the case for Co, Cu chains on Pt(997) do not show evidenceof local magnetic moments, presenting a single 3d feature at 2.3 eV binding energy. No Cu-induced states can be found close to the Fermi level because the Cu 3d shell is filled and thephotoemission cross-section for sp states is very small in this photon energy range. The Cu3d peak shifts to a higher binding energy above 0.17 ML and reaches 2.7 eV at 2.0 ML. Theevolution of the Cu 3d state on Pt(997) also reflects changes in the electronic structure dueto changes in the dimensionality of the system. The observed shift starts around 0.17 ML, incorrespondence with the transition from a 1D to a 2D wire structure. For Cu on Pt(111), wherelarge 2D island growth occurs, the Cu 3d peak is already found at 2.6 eV for coverages lowerthan 0.1 ML, as the atomic coordination already approaches that of a 2D ML [57].

4. Magnetism of Co monatomic chains

The valence band photoemission spectra presented in section 3 suggest the presence ofenhanced S localized on the Co atoms in 1D chains. The magnetic behaviour of 1D wireshas been further studied by XMCD, i.e., the absorption of circularly polarized x-rays in thesoft-x-ray energy range (2p → 3d transitions) [58, 59]. XMCD is defined as a difference inabsorption coefficients for parallel and antiparallel orientation of the helicity of the incidentlight with respect to the magnetization direction of the sample (figure 7). Because of its element

S2540 P Gambardella

770 780 790 800 810

IR - IL + 43˚

IR - IL - 57˚normalized at L2

IR + 43˚

IL + 43˚

IR - 57˚

IL - 57˚

IR - IL - 57˚

Photon Energy (eV)

XM

CD

(a.u

.)X

AS

(a.u

.) z

y

xB

(a)

(b)

Figure 7. (a) Co monatomic chain x-ray absorption spectra for parallel (µ+) and antiparallel(µ−) directions of light polarization and field-induced magnetization (T = 45 K and B = 6 T).The µ+-, µ−-spectra are taken with the field applied in the plane perpendicular to the chains at+43◦ off the (111) terrace normal in the step-up direction (top spectra, black and red (grey) solidcurves, respectively) and −57◦ (bottom spectra, blue (dark grey) and green (light grey) solid curves,respectively). (b) The corresponding dichroic signal (µ+ − µ−) at +43◦ (red (grey) solid curve)and −57◦ (black solid curve). The dashed curve represents the XMCD at −57◦ normalized to theL2 peak intensity at +43◦ .

selectivity and surface sensitivity (down to 3 × 1012 atoms cm−2 [4]), XMCD represents one ofthe most powerful magnetic probes available to date. The technique allows one to identify themagnetization direction and strength and to separately determine S and L for a given elementby means of dipole sum rules [60–62]. The XMCD measurements were performed at beamlineID12B of the European Synchrotron Radiation Facility (ESRF) in Grenoble, where sampleswere prepared in situ at T = 260 K according to the procedure described in section 2. The Cocoverage was controlled by cross-checking the coercive field of Co ultrathin films with thoseobtained at the EPF Lausanne in in situ Kerr–STM experiments. This method is consistentwith the yield of a quartz microbalance and is extremely precise since the coercive field of Coon Pt depends critically on the Co coverage [63]. The absorption spectra were taken in theelectron yield mode by measuring the drain current of the photoexposed sample for parallel andantiparallel alignment of the applied magnetic field B with the light helicity. All spectra havebeen normalized by the photocurrent emitted by a gold mesh positioned before the sample,which serves as a measure of the incident photon flux. The sample was rotated about its polarand azimuthal axes with respect to the incident light direction in order to measure the XMCDalong different crystal orientations, as shown in the inset of figure 7.

Representative results for the XMCD at the Co L2,3 absorption edges for the monatomicwires are shown in figure 7. The amplitude of the dichroic signal is a measure of the

Magnetism in monatomic metal wires S2541

-57˚ +43˚

a) T = 45 K

-4 -2 0 2 4-6 -4 -2 0 2 4 6B (T) B (T)

b) T = 10 K

Mag

netiz

atio

n (a

. u.)

Figure 8. Magnetization of a monatomic wire array as a function of the applied field B. The datapoints represent the Co L3 XMCD at 779 eV normalized by the pre-edge intensity at 775 eV inorder to eliminate the dependence of the electron yield on the sample orientation and magnetic field.(a) Magnetization at T = 45 K, in the easy direction (solid squares, +43◦) and 80◦ away from theeasy direction (empty circles, −57◦) in the plane perpendicular to the wire axis (see the inset). Thedifference between the normalized L3 XMCD at +43◦ and −57◦ at B = 6 T corresponds to thatof the XMCD spectra in figure 7. The solid curves are fits to the data. The dashed curve representsthe magnetization expected for an isolated Co atom on Pt(997). (b) Magnetization at T = 10 K forthe same geometry as in (a). Hysteretic behaviour sets in due to long-range ferromagnetic order.The unsaturated zero-field magnetization is attributed to the inhomogeneous lengths of the chains(from [67]).

magnetization of the Co wire array and contains information on the local character of theatomic moments. Due to the low Co coverage, the absorption edges of the monatomic wiresare superimposed on a strong background originating from the oscillations following the PtN2,3 thresholds. The absorption by the non-magnetic substrate, however, does not contributeto the dichroic effect, as shown by the flat baseline of the Co XMCD in figure 7(b). The wiresare characterized by a strong, angle-dependent dichroism that results from the alignment ofthe Co magnetic moments at B = 6 T, T = 45 K. The easy magnetization direction ofthe monatomic chains lies in the plane perpendicular to the chain axis, tilted by +43◦ withrespect to the (111) surface normal, where the positive sign indicates the step-up direction.The absorption spectra in figure 7(a) taken with the field applied parallel to the easy axis reveala dichroic signal which is more than twice that obtained for the −57◦ direction. In the lattercase, the applied 6 T field is not able to saturate the magnetization, thus indicating the presenceof strong magnetic anisotropy, as discussed later.

The reduced atomic coordination of the monatomic chains compared to the bulk and 2Dfilms has remarkable consequences for the magnitude of S and L. Calculations within thelocal spin density approximation (LSDA) scheme for Co/Pt(997) show that the narrowing ofthe Co 3d band and the corresponding increase in the density of states (DOS) at the Fermilevel (EF ) increment S from the 1.57 µB/atom bulk value to 2.03 and 2.08 µB/atom for aML and a 1D chain, respectively [56]; see also [17–19]. A larger relative increase is expectedfor L, which is generally more sensitive to changes in the atomic coordination [4]. Using theXMCD orbital sum rule [60], L can be experimentally determined from the integrated XMCD,∫

L3+L2(µ+ + µ−) dε = C

2 µBL, where ε is the photon energy and C is an experimental constant

derived from the known bulk value L = 0.15 µB/atom and the bulk integrated XMCD [11].For the monatomic wires we find L = 0.68 ± 0.05 µB/atom, i.e., an enhancement of about afactor 5 compared to hcp Co. The monatomic chain L is significantly larger compared to thatfor 2D multilayers [10, 11, 64, 65], and also to that for 1 ML Co/Pt(111) (L = 0.29 µB/atom).In the 1D chains, the reduced coordination leads to a weaker hybridization of the Co states

S2542 P Gambardella

and, consequently, to the enhancement of the local minority DOS at EF , which leads to anincrease in L [24, 25, 66]. Owing to the Co coordination dependence, larger values of L arefound only in isolated Co adatoms and clusters with up to three atoms [4, 5].

In figure 8(a) we report the magnetic response of a set of monatomic wires at T =45 K. The zero remanent magnetization reveals the absence of long-range ferromagneticorder. However, the shape of the magnetization curve indicates the presence of short-range order, i.e., of significant interatomic exchange coupling in the chains. For non-interacting paramagnetic moments, the magnetization expected in the present experimentalconditions would be significantly smaller, as indicated by the dotted line in figure 8(a). Theobserved behaviour is that of a 1D superparamagnetic system, i.e., a system composed bysegments, or spin blocks, each containing N exchange-coupled Co atoms, whose resultantmagnetization orientation is not stable due to thermal fluctuations. A noticeable dependenceof the magnetization on the direction of the applied field can be observed in figure 8. Thestrongest magnetic response is found in the +43◦ direction, as expected from the XMCDspectra in figure 7. Clearly, the shape of the superparamagnetic curves depends on themagnetic anisotropy energy of each spin block N Ea , as well as on N times the magneticmoment per Co atom. By fitting the curves in figure 8(a) assuming dominant uniaxialanisotropy and a classical model of the magnetization [67], we obtain N = 15 ± 1 andEa = 2.0 ± 0.2 meV/atom. Thus, on average, about 15 Co atoms are coupled in each spinblock at T = 45 K. A simple argument due to Landau [68] shows that this result does notcontradict the spin lattice models treating magnetic order in 1D. Consider a chain consistingof N moments described by the Ising Hamiltonian H = −J

∑N−1i=1 Sz i Sz i+1, with nearest-

neighbour exchange coupling energy J > 0 (ferromagnetic interaction). The ground stateenergy of the system is E0 = −J (N − 1) and corresponds to the situation where all themoments are aligned. The lowest-lying excitations are those in which a single break occurs atany one of the N sites, as shown below:

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ground state,

↑ ↑ ↑ ↑ ↑ ↑ ↑↑ ↑ ↓ ↓ ↓ ↓ ↓ ↓ lowest excited state.

There are N−1 such excited states, all with the same energy E = E0+2J . At temperature T thechange in free energy due to these excitations is �G = 2J − kB T ln(N − 1). For N → ∞ wehave �G < 0 at any finite temperature and the ferromagnetic state becomes unstable againstthermal fluctuations. For (N − 1) < e2J/kB T , however, ferromagnetic order is energeticallystable. Assuming 2J = 15 meV [69, 70], we get an upper limit of N = 50 exchange-coupledatoms at T = 45 K. Measurements of the magnetization in the Co monatomic chains agreewith this limit.

The large magnetic anisotropy energy of the monatomic chains (for comparison, Ea =45 µeV/atom in hcp Co [71]) is directly related [24, 25] to the anisotropy of L along the easyand hard directions. Although the XMCD sum rules cannot be applied far from saturation ofthe magnetization in the hard direction (−57◦), the decrease of the L3 XMCD intensity relativeto L2 (dashed curve in figure 7 (b)) indicates a significant orbital moment anisotropy [5], withL(+43◦) − L(−57◦) ≈ 0.12 µB/atom. As expected, the spin–orbit coupling between L andS favours the direction where L is larger as the easy magnetization axis. The large anisotropyenergy plays a major role in stabilizing long-range ferromagnetic order in 1D, in particular ininhibiting the approach to the thermodynamic limit described above. As in bulk ferromagneticsystems, anisotropy energy barriers can effectively pin the magnetization along a fixed directionin space. By lowering the sample temperature below TB = 15±5 K, we observe a transition toa long-range ferromagnetically ordered state with finite remanence (figure 8(b)). The thresholdtemperature is the so-called blocking temperature, where the magnetization of each spin block

Magnetism in monatomic metal wires S2543

φ

y

Θ

M (

a. u

.)

a) b)

φ (deg)

-90 -60 -30 0 30 60 900.0

0.1

0.2

0.3

0.4

Θ (deg)

-90 -60 -30 0 30 6 00.0

0.1

0.2

z

x

z

Figure 9. The angular dependence of the magnetization near remanence; (a) perpendicular,(b) parallel to the wire axis at 10 K. For each angle the magnetization has been aligned at B = 6 Tand subsequently measured near remanence at 0.25 T. A | cos(x − x0)| dependence is found in bothplanes (solid curves), where x = �, � (from [67]).

aligns along the common easy axis direction and the whole system becomes ferromagnetic.Long-range order in 1D atomic chains therefore enters as a metastable state thanks to slowmagnetic relaxation. According to the Neel–Brown model of magnetization reversal [72],the relaxation time of a single-domain magnetic particle is expressed by an Arrhenius lawof the form τ = τ0 exp( Ea

kT ), where τ0 is a prefactor of the order of 10−9 s. The anisotropyenergy determined for a spin block, N Ea = 31 meV, is thus consistent with TB = 15 Kdetermined from the XMCD data, for which the timescale of the experimental observationrequires τ � 102 s.

As in 2D films, the magnetic anisotropy energy and the easy axis of magnetization aredetermined by the wire atomic structure and by the presence of the substrate. Tight-bindingmagnetic anisotropy energy calculations for free-standing and Pd-deposited Co monatomicchains [16] show that the easy direction rotates from parallel to perpendicular to the chain axisgoing from the free-standing to the Pd-supported chains. Here, the 1D geometry of the wiresand the interaction with the stepped substrate are manifested by a strong uniaxial anisotropicbehaviour and a tilted easy axis with respect to the sample in-plane and out-of-plane directions.Figure 9 shows the projection of the magnetization onto different directions with respect tothe sample normal measured near remanence (B = 0.25 T). As expected for a uniaxialsystem, the magnetization follows a | cos | function both in the plane perpendicular and thatparallel to the wire axis, with a maximum in the perpendicular plane at +43◦. Interactions ofdipolar origin among adjacent chains are far too weak compared to the anisotropy energy ofmagnetocrystalline origin to influence the chain magnetization behaviour [16]. Further, wedid not find evidence for interchain coupling effects mediated by the substrate [73], either offerromagnetic or antiferromagnetic type, which would result in changes of TB with respect tothe value calculated by the Arrhenius law.

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5. Conclusions

One-dimensional models have long been praised for their role in exemplifying and analysingmany-body problems that are common to physics, chemistry, and statistics. It is only recently,however, that 1D systems made of real atoms have become the object of experiments. Arraysof parallel monatomic Co chains can be constructed on a Pt vicinal surface whose steps serveas a deposition template. A narrow temperature range exists where the Co chains grow rowby row. Taking advantage of the uniformity and elevated density of the chain arrays, integralspectroscopic methods can be used to address the electronic and magnetic structure in 1D to 2Dsystems. The development of the Co wire-induced valence band states has been investigatedby means of angle-resolved photoemission spectroscopy as a function of the wire thickness.We found a 3d exchange-split band for the Co chains which indicates the presence of enhancedspin magnetism in 1D. XMCD measurements provided the first experimental insights into themagnetic ordering phenomena of 1D atomic wires. Co monatomic chains sustain both short-and long-range ferromagnetic order depending on the substrate temperature. Owing to slowmagnetic relaxation, ferromagnetic behaviour was observed in the monatomic chains withoutcontradicting thermodynamic restrictions to long-rangemagnetic order in 1D. Ferromagnetismin 1D is stabilized by extraordinarily large magnetic anisotropy energy barriers which arisefrom large, unquenched orbital magnetic moments localized on the Co atoms and hybridizationwith the Pt substrate. Precise control of the atomic coordination in 1D systems represents apromising approach for understanding and tailoring magnetic anisotropy energy barriers innanosized systems.

Acknowledgments

The author would like to acknowledge the people who have contributed to the measurementsand to the discussion of the experimental results presented in this work: M Blanc, K Kuhnke,L Burgi, O Jeandupeaux, H Brune, and K Kern (EPF Lausanne), C Carbone (ConsiglioNazionale delle Ricerche, Trieste, and Forschungszentrum Julich), A Dallmeyer, K Maiti,M C Malagoli, W Eberhardt (Forschungszentrum Julich), O Rader and C Pampuch (BESSYI), P Ohresser, S S Dhesi, N B Brookes, and K Larsson of beamline ID12B at the ESRF.

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